The navy documented a linear relationship between the distance in feet below sea level and the time in seconds traveled by a submarine. What is the rate of change of the distance in feet below sea level with respect to the time that the submarine traveled?

Submarine

Time (seconds)	Distance Below Sea Level (feet)
0	460
18	604
34	732
52	876
70	1020

To determine the rate of change of the distance below sea level with respect to time (in seconds), we need to calculate the slope of the linear relationship between the time and the distance. The slope is given by the formula:

\[
\text{Slope} = \frac{\Delta y}{\Delta x} = \frac{\text{Change in Distance}}{\text{Change in Time}}
\]

Using the data provided:

1. Between the first and second points:
   - \(\Delta \text{Distance} = 604 - 460 = 144\) feet
   - \(\Delta \text{Time} = 18 - 0 = 18\) seconds
   - Slope = \(\frac{144}{18} = 8\) feet per second

2. Between the second and third points:
   - \(\Delta \text{Distance} = 732 - 604 = 128\) feet
   - \(\Delta \text{Time} = 34 - 18 = 16\) seconds
   - Slope = \(\frac{128}{16} = 8\) feet per second

3. Between the third and fourth points:
   - \(\Delta \text{Distance} = 876 - 732 = 144\) feet
   - \(\Delta \text{Time} = 52 - 34 = 18\) seconds
   - Slope = \(\frac{144}{18} = 8\) feet per second

4. Between the fourth and fifth points:
   - \(\Delta \text{Distance} = 1020 - 876 = 144\) feet
   - \(\Delta \text{Time} = 70 - 52 = 18\) seconds
   - Slope = \(\frac{144}{18} = 8\) feet per second

In all cases, the slope is consistent. Therefore, the rate of change of the distance below sea level with respect to time is **8 feet per second**.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you interpret the slope of this linear relationship in a real-world context?
2. What would the distance be after 100 seconds?
3. How can you determine the equation of the line representing this relationship?
4. What would happen if the slope was negative instead of positive?
5. How does the consistency of the slope confirm the linearity of the relationship?

**Tip:** In a linear relationship, the slope represents the rate at which one quantity changes with respect to another.

Discover how to calculate the rate of change of distance below sea level with respect to time for a submarine using linear relationship analysis. Explore step-by-step solutions and understand the slope formula in practical terms. Suitable for high school students.

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